Second Order Circuits
Second order circuits are circuits that have two energy storage elements, resulting in second-order differential equations.
One application of second order circuits is in timing computers. As we will see, an RLC circuit can generate a sinusoidal wave.
There are primarily two types of second order circuits:
- Parallel RLC circuits
- Series RLC circuits
This page will analyze them and derive some useful equations.
Series RLC Circuits
Natural Response
Consider an un-forced RLC circuit. We want to find .
First, we can use KVL and KCL
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle iR + L \frac{di}{dt} + V_C = 0}
Next, we can use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = C \frac{dV_C}{dt}} and substitution to get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RC \frac{dV_C}{dt} + L \frac{d}{dt} \frac{C V_C} {dt} V_C = 0}
Changing the order and moving the constants,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle LC \frac{d^2 V}{dt^2} + RC \frac{dV_C}{dt} + V_C = 0}
Moving constants away from the first term to get a second-order differential equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2V_C}{dt^2} + \frac{R}{L} \frac{dV_C}{dt} + \frac{1}{LC} V_C = 0}
Parallel RLC Circuits
Natural Response
By KCL,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{v}{R} + \frac{1}{L} \int_0^t v d\tau + I_0 + C \frac{dv}{dt} = 0 }
By differentiating once with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} and rearranging some constants,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 v}{dt^2} + \frac{1}{RC} \frac{dv}{dt} + \frac{v}{LC} = 0 }
we get a homogeneous second-order differential equation, which has a standard solution that I will not go into detail. Briefly, it is solved by assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = A e^{st}} since derivatives of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} must take the same form to cancel out to zero.
By applying the standard solution, we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A e^{st} \left(s^2 + \frac{s}{RC} + \frac{1}{LC} \right) = 0 }
Characteristic Equation
The above simplifies to
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^2 + \frac{s}{RC} + \frac{1}{LC} = 0 }
This is the characteristic equation of the differential equation, as the root of the quadratic determines properties of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{1,2} = - \frac{1}{2RC} \pm \sqrt{ \left(\frac{1}{2RC}\right)^2 - \frac{1}{LC}} = - \alpha \pm \sqrt{\alpha^2 - \omega_0^2} }
where
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{1}{2RC} }
and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \frac{1}{\sqrt{LC}} }
It can be pretty easily proven that the sum of the two roots is also a solution
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = A_1 e^{s_1 t} + A_2 e^{s_2 t} }
Forms
Depending on the root, there are three forms:
- Overdamped, where there are real, distinct solutions
- Underdamped, where there are complex solutions
- Critically damped, where the solutions are not distinct.
